In the Figure, ABCD is a rectangle and EFGH is a parallelogram. Using the measurements given in the figure, what is the length d of the segment that is perpendicular to `bar("HE")` and `bar("FG")`?
Solution
In the given figure ABCD is a rectangle and EFGH is a parallelogram.
In the right triangle AEH
HE = `sqrt("AH"^2 + "AE"^2)`
= `sqrt(3^2 + 4^2)`
= `sqrt(9 + 16)`
= `sqrt(25)`
HE = 5
∴ GF = 5 ...(HE and Gf are opposite sides of a parallelogram)
In the right triangle
GC = `sqrt("GF"^2 - "FC"^2)`
= `sqrt(5^2 - 3^2)`
= `sqrt(25 - 9)`
= `sqrt(16)`
∴ DG = 10 – 6 = 4
Area of ΔAEH + Area of ΔBEF + Area of ΔFCG + Area of ΔHDG
= `1/2 xx 3 xx 4 + 1/2 xx 6 xx 5 + 1/2 xx 3 xx 4 + 1/2 xx 5 xx 6`
= (6 + 15 + 6 + 15)
= 42
∴ Area of 4 triangles = 42
Area of the parallelogram = Area of the rectangle ABCD – Area of 4 triangles.
= 10 × 8 – 42
= 80 – 42
= 38
b × h = 38
5 × d = 38
d = `38/5`
= `7 3/5`
Length of d = `7 3/5` or 7.6
